지수적 증가와 감소

The amount of Carbon-14 in the sample $t$ years after the animal died will be $$Q(t)=5e^{-kt}$$ for some constant $k$ (where 5 is the amount of Carbon-14 in the sample at time $t=0$). So, the answer we're looking for is $Q(10000)$. We need to replace $k$ with an actual number to evaluate $Q(10000)$, and the key to doing this is the half-life. The text tells us that the half-life of Carbon-14 is 5730 years, so we know:

$$\begin{aligned} Q(5730)&=\frac{5}{2}\\ 5e^{-k\cdot5730}&=\frac{5}{2}\\ \left(e^{-k}\right)^{5730}&=\frac{1}{2}\\ e^{-k}&=\sqrt[5730]{\frac{1}{2}}=2^{-\tfrac{1}{5730}} \\ \text{So:} \\ Q(t)&=5\left(e^{-k}\right)^t\\ &=5\cdot2^{-\tfrac{t}{5730}} \\ \text{Now, we can evaluate:} \\ Q(10000)&=5\cdot 2^{-\tfrac{10000}{5730}}\approx 1.5~\mu g \end{aligned}$$

Remark: after $2(5730)=11,460$ years, the sample will have been sitting for two half-lives, so its remaining Carbon-14 will be a quarter of its original amount, or $1.25$ $\mu$g. It makes sense that at 10,000 years, the sample will contain slightly more Carbon-14 than at 11,460 years. Indeed, 1.5 is slightly larger than 1.25, so our answer seems plausible.

It's a good habit to look for ways to quickly check whether your answer seems plausible, since a small algebra error can easily turn into a big error in your solution.